Optimal. Leaf size=98 \[ -\frac {5 b^3 \csc (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f} \]
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Rubi [A]
time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2706, 2705,
3856, 2720} \begin {gather*} -\frac {5 b^3 \csc (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2705
Rule 2706
Rule 2720
Rule 3856
Rubi steps
\begin {align*} \int \csc ^2(e+f x) (b \sec (e+f x))^{5/2} \, dx &=\frac {2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {1}{3} \left (5 b^2\right ) \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {5 b^3 \csc (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {1}{6} \left (5 b^2\right ) \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {5 b^3 \csc (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {1}{6} \left (5 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=-\frac {5 b^3 \csc (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 67, normalized size = 0.68 \begin {gather*} \frac {b \left (2-3 \cot ^2(e+f x)+5 \cos ^{\frac {3}{2}}(e+f x) \csc (e+f x) F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.34, size = 202, normalized size = 2.06
method | result | size |
default | \(\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (5 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+5 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-5 \left (\cos ^{2}\left (f x +e \right )\right )+2\right ) \cos \left (f x +e \right ) \left (\cos \left (f x +e \right )+1\right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{3 f \sin \left (f x +e \right )^{5}}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 143, normalized size = 1.46 \begin {gather*} \frac {-5 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (5 \, b^{2} \cos \left (f x + e\right )^{2} - 2 \, b^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{6 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\sin \left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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